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Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays

  • * Corresponding author: Weihua Jiang

    * Corresponding author: Weihua Jiang

The authors are supported by the National Natural Science Foundation of China (No. 11871176)

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  • We consider a two-species Lotka-Volterra competition system with both local and nonlocal intraspecific and interspecific competitions under the homogeneous Neumann condition. Firstly, we obtain conditions for the existence of Hopf, Turing, Turing-Hopf bifurcations and the necessary and sufficient condition that Turing instability occurs in the weak competition case, and find that the strength of nonlocal intraspecific competitions is the key factor for the stability of coexistence equilibrium. Secondly, we derive explicit formulas of normal forms up to order 3 by applying center manifold theory and normal form method, in which we show the difference compared with system without nonlocal terms in calculating coefficients of normal forms. Thirdly, the existence of complex spatiotemporal phenomena, such as the spatial homogeneous periodic orbit, a pair of stable spatial inhomogeneous steady states and a pair of stable spatial inhomogeneous periodic orbits, is rigorously proved by analyzing the amplitude equations. It is shown that suitably strong nonlocal intraspecific competitions and nonlocal delays can result in various coexistence states for the competition system in the weak competition case. Lastly, these complex spatiotemporal patterns are presented in the numerical results.

    Mathematics Subject Classification: Primary: 35B32, 35B35; Secondary: 35B36.

    Citation:

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  • Figure 1.  $ w_{*}>0 $ is a positive root of $ F(w) = 0 $

    Figure 2.  System parameter values are taken as Table 1 shows. The left figure represents the enlarged figure inside the pink rectangular part of the right figure and the red hollow circles are intersection points of $ \mathcal{L}_{n} $ and $ \mathcal{S}_{n} $. The solid black circles means that the remaining curves $ \mathcal{L}_{n} $ and $ \mathcal{S}_{n} $ are omitted here

    Figure 3.  (a), (b) are bifurcation sets and phase portraits respectively

    Figure 4.  When the initial values $ u_{1}(t,x) = 0.45-0.01 $, $ u_{2}(t,x) = 0.27-0.01 $, $ t\in[-\tau,0] $ and parameters $ (\tau,d_{1}) = (\tau_{w_{*}},d_{1}^{1}(d_{2}))+(-0.01,0.01)\in \mathscr{D}_{1} $, the coexistence equilibrium is stable

    Figure 5.  When the initial values $ u_{1}(t,x) = 0.45-0.01 $, $ u_{2}(t,x) = 0.27-0.01 $, $ t\in[-\tau,0] $ and parameters $ (\tau,d_{1}) = (\tau_{w_{*}},d_{1}^{1}(d_{2}))+(0.01,0.01)\in \mathscr{D}_{2} $, a spatial homogeneous periodic orbit is stable

    Figure 6.  For $ (\tau,d_{1}) = (\tau_{w_{*}},d_{1}^{1}(d_{2}))+(0.003,-0.01)\in \mathscr{D}_{4} $, (a), (b) with initial values $ u_{1}(t,x) = 0.45-0.01\cos x $, $ u_{2}(t,x) = 0.27-0.01\cos x $ and (c), (d) with initial values $ u_{1}(t,x) = 0.45+0.01\cos x $, $ u_{2}(t,x) = 0.27+0.01\cos x $, , $ t\in[-\tau,0] $. A pair of spatial inhomogeneous periodic orbits is stable, which indicates $ \mathscr{D}_{4} $ is a bistable region

    Figure 7.  For $ (\tau,d_{1}) = (\tau_{w_{*}},d_{1}^{1}(d_{2}))+(-0.01,-0.01)\in \mathscr{D}_{6} $, (a), (b) with initial values $ u_{1}(t,x) = 0.45-0.01\cos x $, $ u_{2}(t,x) = 0.27-0.01\cos x $ and (c), (d) with initial values $ u_{1}(t,x) = 0.45+0.01\cos x $, $ u_{2}(t,x) = 0.27+0.01\cos x $, , $ t\in[-\tau,0] $. a pair of spatial inhomogeneous steady states is stable, which indicates $ \mathscr{D}_{6} $ is a bistable region

    Table 1.  Values of system parameters

    Parameters $ r_{1} $ $ r_{2} $ $ a_{11} $ $ a_{12} $ $ a_{21} $ $ a_{22} $ $ b_{11} $ $ b_{12} $ $ b_{21} $ $ b_{22} $
    Values $ 5 $ $ 5 $ $ 4 $ $ 7 $ $ 6 $ $ 5 $ $ 3 $ $ 1 $ $ 0.5 $ $ 4 $
     | Show Table
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    Table 2.  Values of system parameters

    Parameters $ d_{2} $ $ r_{1} $ $ r_{2} $ $ a_{11} $ $ a_{12} $ $ a_{21} $ $ a_{22} $ $ b_{11} $ $ b_{12} $ $ b_{21} $ $ b_{22} $
    Values $ 0.6 $ $ 5 $ $ 5 $ $ 4 $ $ 7 $ $ 6 $ $ 5 $ $ 3 $ $ 1 $ $ 0.5 $ $ 4 $
     | Show Table
    DownLoad: CSV
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